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Slide Rules

Linear Slide Rules

Between 1614 and 1622, John Napier discovered logarithms, Edmund Gunter devised a scale on which numerals could be multiplied and divided by measuring the distance between two logarithmic numbers with a pair of dividers, and William Oughtred put two such scales alongside each other, moving one on a slide so that the distance between the numbers could be read off directly. The speed with which these developments unfolded suggests the power that logarithms provided for reducing the tedium of calculation. By the 1680s, the English used slide rules in carpentry and in gauging (estimating liquid volumes)—the instruments were quite helpful for determining excise taxes on barrels of liquor.

It was not until the late eighteenth century that slide rules were commonly utilized in the manufacture of machines and machine tools, most notably by James Boulton and James Watt. Several more decades passed before slide rule use became widespread throughout Europe. French artillery officer Amedée Mannheim fostered acceptance of the rectangular or linear form of the instrument in 1851 by standardizing the general types of scales and the order in which the scales were arranged. Mannheim also developed the cursor or indicator, which made it easier to read results from two scales that were not adjacent to each other. By the 1870s, German companies made slide rules one of the first consumer products to be fashioned out of plastic (specifically, sheets of celluloid laminated to a wooden frame), and they built dividing engines that permitted mass production of the scales engraved on slide rules. In the United States in the 1890s, Keuffel & Esser of New York City pioneered first the import and then the domestic manufacture of linear slide rules. Numerous other companies entered the market in the 20th century.

A quick tour of what you can expect to see on a slide rule starts with the C and D scales, which both represent the standard number line. To multiply two numbers, set the 1 on the C scale above the first number to be multiplied on the D scale. Look at the second number to be multiplied on the C scale; the number below it on the D scale is the answer. For example, to multiply 2 by 3, set 1 on C over 2 on D, and then look below 3 on C to see the answer 6 on D. Division is accomplished by reversing the process. To calculate 6 ÷ 3, set 3 on C over 6 on D, then look at 1 on C to see the answer 2 on D. To deal with numbers larger than 10 or smaller than 1, the user must mentally move the decimal point. Slide rule users also had to be able to estimate distances between marks on the scales, since there was no way to include all of the digits needed to solve a problem such as 3.14 X 2.7. (On linear and circular slide rules, the answer is "approximately 8.48." Web sites on the Resources page provide more detailed training in using slide rules.)

The basic process for setting up and solving problems is the same for operations on other scales. If the numbers used in the calculation produce a result off the ends of the scales, a user employs the CI and DI scales, which put the number line in inverse, or reverse, order. If the numbers still extend past the end of the instrument, the user may try the "folded" CF and DF scales, which start numbering at π instead of at 1. The A and B scales are number lines of squares, so they are used with the C or D scales to square (or take the square root of) a number. The K scale provides cubes and cube roots. L scales represent common logarithms, S scales give sines and cosines, and T scales indicate tangents.

One of the first types of slide rules sold and made in the United States was the carpenter's rule, used for calculations relating to timber, which was one of the country's major early exports. This rule is marked on the lower arm: S. A. JONES & CO. (/) HARTFORD–CON. (/) WARRANTED BOX WOOD. Solomon A. Jones made carpentry tools in Hartford, Conn., from 1838 to 1841. Compare to 1987.0771.01, a British carpenter's rule of the same period. The collections include an image of someone holding the rule (see 2003.0215.02).

This boxwood carpenter’s rule has a brass joint, metal caps at the ends of the arms, and a brass slide. The front of the slide has two identical logarithmic scales labeled B and C. Above is an identical logarithmic scale on the arm; this scale is labeled A. All three of these scales have a double cycle of logarithms, like the A and B scales on a Mannheim rule. Below the C scale on the arm is a scale labeled D and also GIRT LINE, which is divided logarithmically and runs from 4 to 40 (in other words, this is not the D or principal scale of Mannheim slide rules). The girt line can be used with other scales to estimate the volume of timber available from a log.

Underneath the slide is imprinted the number 33. The back of the slide is also stamped with the number 33 and is somewhat crudely marked with a twelve-inch ruler, divided to eighths of an inch. The lower arm contains scales for making scale drawings that are 1/4, 1/2, 3/4, and 1 inch to the foot. The units on the 1/4-inch chain scale appear to be numbered erroneously after 18 (e.g., the next numbered division after 18 is 10 rather than 20). Along the edge is another twelve-inch scale, divided to eighths of an inch.

The other side of the rule has a scale of 24 inches along one edge, divided to sixteenths of an inch. One arm has a scale labeled by twos from 34 to zero and is marked M. A number for 31 is between 32 and 30, and the number for 6 is omitted.

The other arm has a scale marked E and labeled by twos from 24 to zero. The number 24 appears twice, and 6 is omitted. The M and E scales were used in cutting polygonal sections of wood. The outside edge has two scales, each dividing one foot into 100 parts.

By the late 19th century, several American inventors proposed special-purpose slide rules. One of them was James Hogg (1851–1930), an English mechanical engineer who came to the United States in 1880 and soon was overseer at the Globe Worsted Mill in Lawrence, Mass. During his time in Lawrence, Hogg devised this slide rule and arranged to have it manufactured by the Stanley Rule & Level Company of New Britain, Conn. He went on to patent a spindle for spinning machies (U.S. Patent 401,703) and feeding mechanisms for carding machines (U.S. Patents 562,610; 686,071; and 686,072).

This instrument is a wooden linear slide rule with a metal slide and metal edges on all four sides. The scales on the top of the base are lettered A, B, and C; the scales on the slide are lettered D and E; and the scales on the bottom of the base are lettered F, NUM, and LOG. The A and NUM scales range from 1 to 10, divided logarithmically. The B, C, D, and E scales range from 1 to 10 and then from 1 to 10 again, divided logarithmically. The F scale is divided like the B, C, D, and E scales, but runs in the opposite direction. The LOG scale is a scale of equal parts, ranging from 1 to 100 and divided in increments of one-half. The back of the slide is a twelve-inch scale divided to tenths of an inch and to eighths of an inch.

The back of the rule also has a twelve-inch scale divided to tenths of an inch on one edge and a twelve-inch scale divided to eighths of an inch on the other edge. In between are tables providing conversion factors from the volumes of geometric solids to cubic inches and feet; conversion factors from the volumes of geometric solids to the weights in pounds of various substances; the areas of polygons from 5 to 12 sides; the gauge points of a circle; and gauge points for pumping engines, to find the diameters of steam cylinders that will work pumps of specified diameter at 7 pounds per square inch. Compare to 1981.0934.01.

The back is marked: PAT. APPLIED FOR IMPROVED SLIDE RULE (/) ARRANGED & CONSTRUCTED BY (/) JAS. HOGG LAWRENCES MASS. U.S.A. (/) MANUFACTURED BY (/) STANLEY RULE & LEVEL CO. NEW BRITAIN, CONN. No patent has been found for the instrument.

Hogg moved from Lawrence in 1888 and subsequently worked for Stephen Sanford & Sons, Inc., in Amsterdam, N.Y., and the Bigelow Carpet Company in Lowell, Mass. An 1888 advertisement reported that the rule and instruction book sold together for $4.00. The instrument was still on sale in 1908, with instruction book, for a price of $3.50.

This large mahogany linear astronomical slide rule is covered with strips of German silver. There are two slides, each of which have scales on both sides. Each slide has a knob near one end for moving it; these may be unscrewed and attached on the reverse side. One slide is marked COLLIMATION on one side and AZIMUTH on the other. The other slide is marked LEVEL AZIMUTH on one side and REFRACTION on the other. The base has four identical, unlabeled logarithmic scales, each of which runs from 1 to 10 twice (with a bit more at each end).

On the center portion of the base, the instrument is marked Darling Brown & Sharpe Providence R.I. Darling, Brown & Sharpe did business under that name from 1866 to 1892. For additional company history, see 1977.0460.01. According to records of the United States Naval Observatory, this slide rule was purchased for $154.00 in December 1887. Few slide rules specifically for astronomy survive, so these large and expensive objects were probably not widely used. Compare to two late 19th-century rules held by the Powerhouse Museum, http://www.powerhousemuseum.com/collection/database/?irn=233017.

Reference: Ledger of Instruments Purchased by the U.S. Naval Observatory, ca. 1845–1906, United States Naval Observatory, Washington, D.C.

This ten-inch mahogany rule is coated with white celluloid on the front and back and is held together with straight brass end pieces. Two screws are in one of the end pieces. The indicator is brass in the double chisel style. On one side, the bottom of the base is marked: WM COX. PAT. OCT. 6. 1891. The right side of the top of the base and the slide are both marked: *. On the other side, the bottom of the base is marked: KEUFFEL & ESSER Co. NEW-YORK.

The scales are not labeled. On both sides, the top of the base has an A scale (divided logarithmically from 1 to 10 twice) and the bottom of the base has a D scale (divided logarithmically from 1 to 10 once). One side of the slide has B and C scales. (A B scale is equivalent to an A scale, and a C scale is equivalent to a D scale.) The other side of the slide has BI and CI scales (i.e., B-inverse and C-inverse).

British citizen William Cox received U.S. patent number 460,930 for a duplex "Engineer's Slide-Rule" with these scales on October 6, 1891, and immediately assigned the patent to the Keuffel & Esser Company of New York. This slide rule sold as K&E model 1744 from 1895 to 1900 and as model 4070 from 1901 to 1906. In the 1895 K&E Catalog, which includes the patent drawings in the illustration, the K&E mark is shown near the top of the base instead of near the bottom of the base. The mark is shown near the bottom of the base in the 1897 and 1899 catalogs. The shape of the end pieces was altered in 1901, and a second style of indicator, the glass clam shell, was added as model 4071. By 1909, K&E discontinued model 4070. Thus, this example likely dates to 1896–1900. It would have been partially made by Dennert & Pape in Germany and finished in the United States by K&E, which sold it in a case for $6.50.

K&E donated this slide rule to the Smithsonian in 1961. Compare to model 4071: MA*318475, MA*326613, and 1993.0482.01.

This ten-inch wood slide rule is faced with white celluloid on both sides. The indentations on the metal end pieces face in, as was typical for Keuffel & Esser slide rules manufactured between 1901 and 1913; compare to MA*326613. The brass indicator is in the double chisel style (with an open interior), used from 1891 to about 1909. See also MA*318477 and 1977.0370.01. The edge of the indicator is marked: KEUFFEL & ESSER CO. (/) NEW-YORK.

The slide has two parts, which fit between the three parts of the base. On one side, the top of the base has an A scale; the top of the slide has B and C scales; the middle of the base has D and DF scales; the bottom of the slide has C and S scales; and the bottom of the base has an A scale. The bottom of the base is marked: KEUFFEL & ESSER Co NEW-YORK. On the left edge of the rule, the top and bottom of the base and the slide are all stamped: 5.

On the other side, the top of the base has an A scale; the top of the slide has BI and CI scales; the middle of the base has D and DF scales; the bottom of the slide has CIF, L, and T scales; and the bottom of the base has a D scale. On both sides, the DF scale is unlettered. The CIF scale is incorrectly lettered "CI." Unlike later slide rules, which start at pi, the folded scales on this rule (CF, CIF, and DF) start at the square root of 10 (approximately 3.16).

The bottom of the base on the back is marked in red: PAT. OCT. 6. 1891 & MAR. 21. 1899. The second patent date is in a different font and size than the first. The first patent refers to William Cox's invention of the duplex slide rule, which K&E used to launch its manufacture of American slide rules. William L. E. Keuffel, the company's head of manufacturing, received the second patent for the design of the dual slide seen in this rule.

K&E only used the dual slide on its Universal slide rule, suggesting the concept was not as useful as the company hoped. The dual slide also addressed a limitation of K&E dividing engines, which were only able to cut scales on the edges of rules. Once K&E improved its dividing engines, this solution was no longer necessary. The Universal slide rule sold as model 4090 from 1900 to 1905 and as the model 4090N from 1906 to 1911. From 1900 to 1902, it had a brass chisel indicator; from 1903 to 1905, it had a German silver chisel indicator; and from 1906 to 1911 it had a glass clam shell indicator. This example thus dates to 1900–1902. It sold for $20.00.

This 20-inch mahogany linear slide rule is coated on the front and back with white celluloid; the edges are bare. The scales are labeled on the right end of each side. On both sides, the top of the base has an A scale, and the bottom of the base has a D scale. On one side, the slide has B and C scales; on the other, the slide has BI and CI scales. The rule has two indicators: the original brass double-chisel style indicator, and the frameless glass with plastic edges that Keuffel & Esser adopted in 1915. According to the donor, the second indicator was acquired some years after the original slide rule.

The bottom of the base on one side and the chisel indicator on the other side are both marked: KEUFFEL & ESSER Co. NEW-YORK. The bottom of the base is also engraved: CHAS. C. BRUSH 1901. The symbol for pi has also been scratched on the D scale on that side. On the other side, the bottom of the base is marked: PAT. OCT. 6. 1891. One edge of the frameless indicator is marked: K&E.CO.N.Y. (/) PAT.8.17.15. For information on these patents, see MA*318477 and MA*318475.

The cardboard case is covered with brown morocco leather. It is marked: KEUFFEL & ESSER CO. (/) DUPLEX (/) SLIDE RULE. See the similar case for MA*326613.

According to the donor, the instrument was used by his father, Charles C. Brush (1880–1968), who graduated from high school in Philadelphia in 1898 and subsequently studied at the Franklin Institute. He received a certificate from the School of Naval Architecture there in 1904. The model 4078 is only listed in Keuffel & Esser catalogues for 1901 and 1903, priced at $16.50. The catalog evidence and the date on the slide rule make it reasonable to suppose that Brush acquired the slide rule in the course of his studies. Charles C. Brush was then associated with the Bureau of Lighthouses from 1917 to 1939. When the Lighthouse Service ended in 1939, he served as a marine engineer in the engineering department of the U.S. Coast Guard until his retirement in 1944.

This ten-inch duplex mahogany slide rule is coated with white celluloid on the front and back; the edges are bare. Unlike earlier slide rules from the Keuffel & Esser Company, such as MA*318477, the end pieces are silver-colored metal (not brass) and the indented edges of the end pieces face in instead of out. The indicator is glass in a brass frame, which was called the "clam shell" style.

The scales are labeled. On both sides, the top of the base has an A scale and the bottom of the base has a D scale. The slide has B and C scales on one side and BI and CI scales on the other side. On one side, the bottom of the base is marked: KEUFFEL & ESSER Co.N.Y. It is also marked there: B.S.5470. On the other side, the bottom of the base is marked in red: PAT. OCT. 6. 1891. The bottom edge of the indicator is engraved: KEUFFEL & ESSER CO (/) NEW–YORK.

The instrument fits in a cardboard box covered with black morocco leather. The large part of the box is embossed: KEUFFEL & ESSER Co. (/) DUPLEX (/) SLIDE RULE. The U.S. National Bureau of Standards (now the National Institute of Standards and Technology) transferred this instrument to the Smithsonian in 1965 as excess property.

K&E sold this instrument as model 4071 from 1901 to 1917. By this time, K&E was manufacturing slide rules from start to finish at its factory in Hoboken, N.J. In 1913, the shape of the end pieces was changed to an "L," as with MA*318475. That instrument also has patent dates of 1900 and 1908 on it, suggesting that this slide rule dates from 1908 or earlier. The 1909 catalog describes the 1908 patent as "pending." Model 4071 initially sold for $8.00 and was reduced to $7.00 in 1906 and to $5.00 in 1909.

This 22-inch, two-sided wooden slide rule has scales that are printed on paper but not engine-divided. On the front of the base, logarithmic scales are labeled B and C on the left and D and D on the right. The C scale is the inverse of the B scale and is also labeled RECIPROCALS. The first D scale runs from 1 to 3.16, and the second D scale runs from 3.16 to 10. These scales are labeled ROOTS.

Four identical scales are on the slide, all logarithmically divided from 1 to 10 and labeled A. Two of the scales are at the left end of the slide, and two are at the right end of the slide. These scales are also identical to the B scale. Written in pencil on the back of the slide and underneath the slide is the number 1103. The bottom of the base is marked: Patented July 2, 1901.

The back of the base has scales divided logarithmically from 1 to 10 and from 3.16 to 10 to 3.16. These scales are both labeled F at both ends. The slide also has scales divided logarithmically from 1 to 10 and from 3.16 to 10 to 3.16. These scales are both labeled E at both ends. Faded red numbers from 1 to 4 also appear on all four scales. Written in pencil on the back of the slide and underneath the slide is: 1102. The bottom of the base is marked: Patented July 2, 1901.

The top edge of the rule is marked: THE ENGINEERS (/) SLIDE RULE. It is also marked: From E. M. Scofield, Bridge Engineer. (/) Sta. A, – Youngstown, – Ohio. (/) Rule sent prepaid on receipt of price. (/) Factory, – Youngstown, Ohio. The paper pasted on this edge also has the formula ex/f, a description of how to solve this formula with the instrument, and tables of equivalents. The bottom edge of the rule gives seventeen more formulas and instructions. It is marked: DIRECTIONS FOR SCALES A • B • C & D (/) FOR "Op." READ "OPPOSITE" (/) FOR "—" READ "ON SCALE" (/) FOR "Ans." READ "FIND ANSWER" (/) FOR LAST SIX PROBLEMS REVERSE SLIDE.

Edson Mason Scofield (1867–1939) developed this rule in 1891 while he worked for Edwin Thacher (inventor of the cylindrical slide rule bearing his name; see, for example, MA*312866). Thacher applied for a patent on the design in 1900 and assigned a half-interest to Scofield when the patent was issued in 1901. According to the instrument, Scofield may have distributed the rule himself.

It was also sold by Eugene Dietzgen Company of Chicago from about 1901 through at least 1931. The price for model 1787 ranged from $5.00 to $7.00. By 1926 Dietzgen marked the rule with its company name. Meanwhile, Scofield went on to become president and chief engineer of the National Bridge Company of Pittsburgh in 1901. From 1903 he and his brother, Glenn M. Scofield, operated the Scofield Engineering Company in Philadelphia, New York City, and Los Angeles. This example belonged to Robert A. Cummings, a civil engineer who was the father of the donor. Compare to MA*333710.

The scales on this 22-inch, two-sided wooden slide rule are printed on paper. The front has two scales on the left end of the base labeled B and C and two scales labeled A on the left end of the slide. These scales run half the length of the instrument. The B and two A scales are identical, divided logarithmically from left to right and running from 1 to 10. The C scale is divided logarithmically from right to left and runs from 10 to 1. It is labeled RECIPROCALS.

The right end of the base has two D scales, both labeled ROOTS, with two additional A scales on the right end of the slide. The A scales are divided logarithmically from left to right and run from 1 to 10. The top D scale runs from 1 to 3.16 and shows square roots of the numbers on the A scale below it. The bottom D scale runs from 3.16 to 10 and shows square roots of the numbers on the A scale above it, between 10 and 100. Written in pencil on the back of the slide and underneath the slide is the number 1096.

The back of the rule contains two scales on the base labeled F and two scales on the slide labeled E. The top F scale runs the length of the scale and is divided logarithmically from 1 to 10. The sections between 1 and 2, between 2 and 3, between 3 and 4, and between 4 and 5 are subdivided. The lower F scale is similarly divided, but starts from the center of the rule (running from the square root of 10, or 3.16, to 10 and then from 1 to the square root of 10). The E scales are identical to the adjacent F scales. Written in pencil on the back of the slide and underneath the slide is the number 1097.

The bottom of the base on both sides is marked: Patented July 2, 1901. The top edge of the rule is marked: THE ENGINEERS (/) SLIDE RULE. It is also marked: From E. M. Scofield, Bridge Engineer. (/) Sta. A, – Youngstown, – Ohio. (/) Rule sent prepaid on receipt of price. (/) Factory, – Youngstown, Ohio. The paper pasted on this edge also has the formula ex/f, a description of how to solve this formula with the instrument, and tables of equivalents. The bottom edge of the rule gives seventeen more formulas and instructions. It is marked: DIRECTIONS FOR SCALES A • B • C & D (/) FOR "Op." READ "OPPOSITE" (/) FOR "—" READ "ON SCALE" (/) FOR "Ans." READ "FIND ANSWER" (/) FOR LAST SIX PROBLEMS REVERSE SLIDE.

Edson Mason Scofield (1867–1939) developed this rule in 1891, while he worked for Edwin Thacher (inventor of the cylindrical slide rule bearing his name; see, for example, MA*312866). Thacher applied for a patent on the design in 1900 and assigned a half-interest to Scofield when the patent was issued in 1901. According to the instrument, Scofield may have distributed the rule himself. It was also sold by Eugene Dietzgen Company of Chicago from about 1901 through at least 1931. The price for model 1787 ranged from $5.00 to $7.00. By 1926, Dietzgen marked the rule with its company name. This instrument is identical to MA*325990.

John N. Welsh (1904–1999), the donor of this slide rule, was a chemical engineer who obtained his B.S. from the Carnegie Institute of Technology in 1925. He was associated with Hall Laboratories, Inc. of Pittsburgh (later CALGON) from 1939 through at least 1968. By 1972, when this slide rule was briefly placed on exhibit, Welsh was living in Florida.

Edward Wager-Smith (1872–1920), a native of New Jersey, graduated from the Spring Garden Institute in Philadelphia in 1889 and in 1893 gained employment as a draftsman for Merritt & Company, a structural steel firm in Philadelphia. As he rose to the position of structural engineer by 1910, he invented the Wager Timber Scale and the Merritt Beam Scale (1987.0108.02). This rule assisted architects and construction workers with computing the strength of wooden beams. It has 11 scales: A, thickness in inches; B, depth in inches; C, spacing in feet; D, span in feet; E, fibre [sic] stress in pounds per square inch; F, load in pounds per square foot; G, type of wood; H, depth in inches; I, method of loading; J, span in feet; and K, fibre stress in pounds per square inch. Scales A-B-C-D-E-F are meant to be used together, as are scales A-B-I-J-K-L and scales G-H-I-J.

The back of the instrument gives government recommendations and regulations for safe stresses on wooden beams and a moisture classification chart by A. L. Johnson, "Economical Designing of Timber Trestle Bridges," U.S. Department of Agriculture Division of Forestry Bulletin No. 12 (Washington, D.C.: Government Printing Office, 1902), 11–12. The instrument fits in an orange paper envelope, which has instructions on its flap.

This rule was distributed by the John Howard Herrick Company of Baltimore, a dealer of hardware and building materials. A photographer by avocation, Wager-Smith also designed the Wager Exposure Scale (patented in 1901—see 1993.0386.01) for correctly timing photographic exposures and the Wager Definition Scale (copyright 1905—see 1993.0386.02 and PG*4750) for determining the longest exposure allowable to produce clearly defined images of moving objects.

The instrument is marked on the front and on the envelope: The Wager Timber Scale (/) FOR COMPUTING THE STRENGTH OF WOODEN BEAMS. It is also marked in both places: THE JOHN HOWARD HERRICK CO. (/) BALTIMORE, MD., U.S.A. It is also marked in both places: PRICE ONE DOLLAR. It is also marked on the front: PAT. JULY 1ST, 1902. This refers to a patent for a "slide-scale" resembling the Merritt Beam Scale and issued on that date to James S. Merritt of Philadelphia. The Wager Timber Scale was advertised for sale from Philadelphia by January 1905 and was mentioned in a textbook as late as 1921.